The overall goal is to optimize the design of medical imaging systems and reconstruction algorithms for the purposes of tumor detection in humans and animals. For imaging systems this is done by devising efficient methods to calculate the performance of ideal observers that use the noisy data from the system on realistic tumor detections tasks. These methods make use of symmetries of the imaging system, consistency conditions on the data, and constraints on the objects to simplify the computations. Noise models are chosen to reflect the measurement noise in the system, the background variation in the patient population, and the normal variation in tumor characteristics. Performance of ideal observers is determined by the area under the receiver operating characteristic curve. For a given detection task, this performance is a function of the parameters in the system design. By varying these parameters the optimal design for tumor detection is found by maximizing this figure of merit. When the ideal observer performance is too difficult to compute, the ideal linear observer is substituted. For reconstruction algorithms, the performance of linear mathematical observers using the reconstructed images for tumor detection tasks is computed. These model observers are chosen to match the performance of human observers on similar tasks. To compute the performance of these observers, the first-order and second-order statistics of the reconstructed images are calculated or approximated. These statistical moments are then used to compute the signal-to-noise ratio for the model observer on the given tumor detection task. Of particular interest is the role of redundant data and null functions of the imaging system in the deterministic and statistical properties of the reconstruction algorithms.